Newton s Laws of Motion & Gravity 23 Sept. First test
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1 Newton s Laws of Motion & Gravity 23 Sept De Revolutionibus Orbium Coelestium, Copernicus, 1543 Astronomia Nova, Kepler, 1609 Philosophiae Naturalis Principia Mathematica, Newton, 1687 Newton: Same laws apply to a falling apple & moving planet. Description of motion Newton s Law of Gravity (Force 1/R 2 ) implies Kepler s 3 rd Law Isaac Newton (at 47) by Godfrey Kneller Trustees of the Portsmouth Estate Copernicus Columbus sails Tycho Brahe Shakespeare Johannes Kepler Jamestown King James Bible Harvard College Isaac Newton George Washington First test See practice test (link is on the syllabus) A few questions with verbal, numeric, or graphical answers. No multiple-choice questions. Material covered today will be on the test. First test is low risk: it counts only 5% of course grade. Homework 3 must be handed by noon, Tues, 29 th. Answers will be posted afterwards. See link on syllabus. Class on Mon is Missouri Club You must ask a question, preferably a question of detail. How do you do question 3 is not detailed enough. You may bring one sheet of notes to use for Test 1. 8½ 11. Write on front & back. 1
2 Nature and Nature's laws lay hid in night./ God said, Let Newton be! and all was light. Newton s epitaph by Alexander Pope Natural Motion for Newton & Aristotle Natural motion is motion that needs no explanation: the object naturally moves that way. Aristotle: For heavenly objects, natural motion is motion in a circle with the same speed. For base objects, natural motion is rest. A book falls off the table and comes to rest on the floor. This needs no explanation because rest is the natural state. Newton: Natural motion is moving at the same speed in the same direction. Newton s First Law: In the absence of a force, an object moves at the same speed in the same direction. 1. A book falls off the table and lands on the floor. For Newton, what is natural, needing no further explanation? a. The book is on the floor. b. The book is halfway to the floor. c. The book is just starting to fall. 2
3 Natural Motion for Newton & Aristotle Natural motion is motion that needs no explanation: the object naturally moves that way. Aristotle: For heavenly objects, natural motion is motion in a circle with the same speed. For base objects, natural motion is rest. A book falls off the table and comes to rest on the floor. This needs no explanation because rest is the natural state. Newton: Natural motion is moving at the same speed in the same direction. Newton s First Law: In the absence of a force, an object moves at the same speed in the same direction. 2. Venus moves around the sun in a circle at the same speed. Does Newton consider this motion natural? a. Yes, Venus is a heavenly object. b. Yes, the speed is the same. c. No, the direction is not always the same. d. No, Venus is not at rest. Newton s Second Law Newton s First Law: In the absence of a force, an object moves at the same speed in the same direction. Velocity is the combination of speed and direction of motion Specify speed and direction: I drive 10 north of east at 50mph. Draw an arrow. Length specifies speed 3
4 Change in velocity Q The velocity changed in a. Case A only b. Case B only c. Neither cases A nor B d. Both cases A and B Case A Case B Case C Case D Velocity after 2 s Acceleration Q The velocity changed in a. Case A only b. Case B only c. Neither cases A nor B d. Both cases A and B Acceleration is change in velocity divided by amount of time Acceleration is arrow from the tip of the beginning velocity to the tip of the ending velocity divided by time Draw the acceleration for cases C & D. The length of the acceleration is greatest for which case? A, B, C, or D Case A Case B Case C Case D Velocity after 2 s 4
5 Newton discovers the law of gravity Newton was sitting under an apple tree and looking at the moon. An apple falls on his head. Newton realizes the moon and the apple fall for the same reason. He does a quick calculation and concludes that the force of gravity decreases as the square of the distance. In a second, an apple falls 5 m. The moon stays up in the sky. What does Newton mean by the moon falls? The moon is 60 times farther from the center of the earth than the apple. It falls 1/60 2 as far as the apple. 1.4mm Moon, 1 s later Moon 5m/(1.4mm)=3600=60 2 If moon fell 83mm, then force of gravity falls as distance 5m/(83mm)=60 Earth 5 m Newton Implies Kepler s 3rd Law Easier derivation: Assume orbit is a circle. Ignore numerical constants such as π or 2. Newton s Law of Gravity: Force between sun and planet Force = G mass Sun mass Planet /Distance 2 ; F = G M m/r 2 Newton s 2 nd Law Force = mass Planet acceleration; F = m a G M m/r 2 = m a; mass of planet cancels out. Velocity is approximately R/P, where P is period. (It is exactly 2πR/P.) Acceleration, change in velocity/time, is approximately (R/P)/P. G M /R 2 = a =R/P 2 P 2 = R 3 /(G M) More accurate derivation P 2 = 4 π 2 /G R 3 /(M sun +m planet ) 4π 2 /G = 2x10 30 kg yr 2 /AU 3 = 1 M sun yr 2 /AU 3 5
6 Kepler s 3 rd Law, according to Newton If period is measured in years, the semi-major axis is measured in AU, and mass of star is measured in M sun, and mass of planet is negligible, P 2 = R 3 /M star Kepler s 3 rd Law is the special case of a planet with negligible mass in orbit around a star with the mass of the sun. 1. A planet orbits a star with the same orbit as the earth. Its period is 2 years. The mass of the star is M sun. A. 2 B. 4 C. ½ D. 1/4 Summarizing Questions What did Newton learn about Kepler s 3 rd law that Kepler did not know? Does P 2 = R 3 /M, where P is the period in years and R is semi-major axis in AU apply to the Hubble Space Telescope in orbit around Earth? A. No, B. Yes What is the reason? 6
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